Applications of Laguerre geometry in CAGD

Helmut Pottmann; Martin Peternell

doi:10.1016/S0167-8396(97)00023-X

Applications of Laguerre geometry in CAGD

Helmut Pottmann^*, Martin Peternell

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

96 Scopus citations

Abstract

We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersections and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in ℝ⁴. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.

Original language	English (US)
Pages (from-to)	165-186
Number of pages	22
Journal	Computer Aided Geometric Design
Volume	15
Issue number	2
DOIs	https://doi.org/10.1016/S0167-8396(97)00023-X
State	Published - Feb 1998
Externally published	Yes

Keywords

Blending
Canal surface
Dupin cyclide
Geometrical optics
Laguerre geometry
Medial axis
NC milling
Rational curve
Rational surface
Voronoi curve
Voronoi surface

ASJC Scopus subject areas

Modeling and Simulation
Automotive Engineering
Aerospace Engineering
Computer Graphics and Computer-Aided Design

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10.1016/S0167-8396(97)00023-X

Cite this

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title = "Applications of Laguerre geometry in CAGD",

abstract = "We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersections and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in ℝ4. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.",

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year = "1998",

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doi = "10.1016/S0167-8396(97)00023-X",

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AU - Pottmann, Helmut

AU - Peternell, Martin

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AB - We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersections and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in ℝ4. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.

KW - Blending

KW - Canal surface

KW - Dupin cyclide

KW - Geometrical optics

KW - Laguerre geometry

KW - Medial axis

KW - NC milling

KW - Rational curve

KW - Rational surface

KW - Voronoi curve

KW - Voronoi surface

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DO - 10.1016/S0167-8396(97)00023-X

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EP - 186

JO - Computer Aided Geometric Design

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Applications of Laguerre geometry in CAGD

Abstract

Keywords

ASJC Scopus subject areas

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